Cardiovascular System

Abstract

The components and operation of the human circulatory system are overviewed, followed by a detailed discussion about the properties, pressure, and flow of blood in it. This leads to analysis of the physical aspects of strokes and aneurysms. Models of the circulatory and systems are developed and applied to the normal and malfunctioning heart.

Problems

Blood Pressure

8.1

(a) The brain in a human is 55 cm above the heart. If the average blood pressure in the major arteries near the heart is 100 mmHg, what is the blood pressure in major arteries in the brain (in both mmHg and cmH$_{2}$O) when a person is either lying down or standing up?

(b) Repeat part (a) for an erect human on the moon $(g=g_{\mathrm {Earth}}/6)$ and on Jupiter $(g=2.34g_{\mathrm {Earth}})$.

(c) A pilot coming out of a dive experiences an upward centripetal acceleration a of magnitude $v^{2}/r$, where v is the speed of the jet and r is the radius of curvature of the trajectory, that adds to gravity (effectively increasing g to $g+a$). What is the arterial pressure in the pilot’s brain for v = 200 m/s and $r = 2$ km? What could happen to the pilot during this recovery from the dive? Would you expect dizziness because of a lack of blood to the head? (See Fig. 8.63.)

(d) What must the pressure in the aorta in a giraffe be (on Earth) for its brain to receive blood? (How can you estimate the elevation of its brain above its aorta?)

Fig. 8.63

Trajectory of a pilot coming out of a dive, with the acceleration and velocity vectors shown for the low point of the dive. For Problem 8.1 (From [4])

8.2

In Problem 8.1d we saw that the pressure of the blood leaving the heart of a giraffe and entering its systemic system must be much larger than that for humans because of its long neck.

(a) Would a blood pressure of 280 mmHg/180 mmHg account for pumping the blood up this long neck in large arteries and then for the pressure drop that occurs in the very small arteries in the brain?

(b) The giraffe has this relatively high blood pressure because of this long neck, which is usually nearly vertical. However, we could expect that there would be a rush of blood to the brain because of this high pressure when the giraffe lowers its head by almost 7 m when it bends down to drink water, and that this could lead to rupture of the arteries in the brain. Why?

(c) Such artery rupturing does not occur because of compensating effects. Explain this by considering the following: (i) The elastic walls of the long giraffe carotid artery help force blood upward (which is a peristaltic action), and this also means that this artery can swell to absorb excess blood when the head is lowered because it is very compliant. (ii) The giraffe jugular vein contains a series of one-way valves that prevent back-flow of the blood when the giraffe’s head is down.

8.3

When you stand on your head, why does your head become red and why do your legs become pale?

8.4

Why is blood pressure measured using major arteries in the upper arm, rather than those in the lower arm or leg?

8.5

You are told that your blood pressure is 880 mmHg/840 mmHg. You are quite understandably concerned because these values are astronomically high, but you are told not to worry because your blood pressure is normal. Should you be concerned?

8.6

(a) An intravenous infusion is made under gravity. If the fluid to be delivered has a density of 1.0 g/cm$^{3}$, at what height above the vein, h, should the top surface of the fluid in the bottle be positioned so the fluid just barely enters the vein? The gauge pressure in the vein is 18 mmHg. (Assume the needle entering the vein has a “large” inside diameter.) (See Fig. 8.64.)

(b) If this needle has a “very small” inside diameter, should the bottle be placed higher, lower, or at the same height? Why?

(c) Why are such infusions performed intravenously and not intra-arterially?

8.7

You are lying down and are injured in such a way that blood from a major artery squirts upward. How high can it spurt?

8.8

Twirl one arm around as fast as you can many times until you see your fingers on that arm turn red. (Continue even if you do not see them getting redder.) [2]

(a) Estimate the centrifugal acceleration at the end of your finger tips, $v_{\mathrm {radial}}^{2}/r=v_{\mathrm {radial}}^{2}/l_{\mathrm {arm}}$, where $l_{ \mathrm {arm}}$ is your arm length (to your fingertips) and $v_{\mathrm {radial}}$ is the radial speed of your finger tips. (Why is $v_{\mathrm {radial}} = 2 \pi l_{\mathrm {arm}}/T$, where T is the period for a complete cycle of this motion?)

(b) Express this acceleration in units of g.

(c) Calculate the effective pressure pushing your blood to your fingers by this motion. This is the apparent outward force divided by the cross-sectional area of your fingers, $A_{\mathrm {fingers}}$, or $(m_{\mathrm { fingers}}v_{\mathrm {radial}}^{2}/l_{\mathrm {arm}})/A_{\mathrm {fingers}}$. Because $m_{\mathrm {fingers}}=$ $\rho _{\mathrm {fingers}}l_{\mathrm {fingers}}A_{ \mathrm {fingers}}$, this pressure is $\rho _{\mathrm {fingers}}v_{\mathrm {radial} }^{2}(l_{\mathrm {fingers}}/l_{\mathrm {arm}})$.

(d) Compare this to the Systolic pressuresystolic pressure 120 mmHg and explain why your fingers (could have or should have) turned red.

Fig. 8.64

Intravenous infusion under gravity. (Based on [26].) For Problem 8.6

Blood Pressure Drop During Flow

8.9

For smaller arteries, how much larger is the total cross-sectional area of the two daughter vessels thanPoiseuille’s Law the parent?

8.10

Calculate the pressure drop (in mmHg) across the following arterial systems using Poiseuille’s LawPoiseuille’s Law with $\eta _{\mathrm {blood}} = 4 \times 10^{-3}$ Pa-s, for a total flow of 80 cm$^{3}/$s across each system:

(a) aorta (internal radius $r = 1.25$ cm, length $L = 10$ cm, all of the flow in this one aorta)

(b) large arteries ($r = 0.2$ cm, $L = 75$ cm, $n = 200$ of them, each with equal flow and the same dimensions)

(c) arterioles ($r = 30\,\upmu$m, $L= 0.6$ cm, $n = 5 \times 10^{5}$)

(d) capillaries ($r = 3.5\,\upmu$m, $L = 2$ mm, $n = 10^{10}$).

8.11

In estimating pressure drops across the different arterial branches we assumed specific numbers of arteries of given diameters and lengths. There is really a wide range of arterial diameters and lengths. How does this affect the pressure drops in the systemic arterial system?

8.12

In estimating pressure drops across the arterioles we assumed a specific number of arterioles with the same diameter and length.

(a) Let us say that all arterioles have the same radius, but their lengths (instead of all being L) range between 0.8L and 1.2L (with equal probability throughout). How does this change the overall resistance of the arteriole system?

(b) Let us say that all arterioles have the same length, but their radii (instead of all being r) range between 0.8r and 1.2r (with equal probability throughout). How does this change the overall resistance of the arteriole system?

(c) The pressure drop across each arteriole in the system must be the same because each is fed by the large arteries, whose pressure is set by the left ventricle, and by the arterial side of the capillaries, whose pressure is also set. If the overall pressure drop across the arterioles is unchanged (by the changes in (a) or (b)), how is the overall flow rate in the arteriole system changed and what is the flow in each arteriole, for the situations alternately described in (a) and (b)?

(d) If you wanted the flow rate to stay the same in each arteriole in (a) and (b) for the given distributions of lengths and radii, how would you have to change the distributions of radii and lengths in each, respectively?

8.13

Find the pressure drop across the arterioles in Problem 8.10c, if—with the same total flow in both cases—and either

(a) all the arterioles become clogged in such a way that their radii decrease to $28\,\upmu$m or

(b) the number of the arterioles decreases to $4 \times 10^{5}$.

(c) By how much would the pressure in the main arteries need to change if the body responded to either change by maintaining the same flow rate?

8.14

Assume that the diameter of each blood vessel in a person is doubled and the total volumetric flow rate is not changed.

(a) What is the new total volume of blood? (Assume the base line parameters in the chapter.)

(b) What is the new circulation time for blood (which is the total blood volume/total volumetric flow rate)?

(c) How do the resistances of the arterioles and capillaries change?

(d) How do the pressure drops across the arterioles and capillaries change?

(e) How does the work done by the heart change?

8.15

Repeat Problem 8.14 if instead the length of each blood vessel is doubled.

8.16

The length of a blood vessel is doubled and its diameter is doubled.

(a) How does the flow resistance change?

(b) If the flow through it is unchanged, how does the pressure drop change?

(c) How does the flow through it change if instead the pressure drop is unchanged?

8.17

Your internal body temperature increases from 37 to 40 $^{\circ }$C. Assuming that the only thing that changes is the viscosity of blood, how must the blood pressure change to ensure the flow rate remains unchanged?

8.18

Poiseuille’s LawUse (8.20) to relate P(L) to P(0) and other flow terms.

8.19

(a) Poiseuille’s LawEstimate how much the flow is changed in small arteries by including the influence of compliance by using (8.20) and assuming the same pressure drop.

(b) Repeat this estimate for how much the pressure drop changes with this analysis assuming the same flow rate.

8.20

Express the flow resistance units of N-s/m$^{5}$ in terms of (N/m$^{2}$)/(cm$^{3}$/s), dyne-s/cm$^{5}$, and PRU (with 1 PRU = 1 mmHg-s/mL).

8.21

(a) Calculate the total peripheral vascular resistance in the systemic and pulmonary systems for someone with a steady-state blood flow rate of 5 L/min, and with 120 mmHg/80 mmHg blood pressure in the systemic system and 25 mmHg/8 mmHg pressure in the pulmonary system. Express your answer in the units of dyne-s/cm$^{5}$, which are CGS units and those that are often used by cardiologists.

(b) The expression given in the text for this vascular flow resistance should be corrected because it uses the average pressure at the beginning of the system instead of the pressure drop across the system. For the systemic system the final pressure is that at the right atrium (2 mmHg) and for the pulmonary system it is that at the left atrium (5 mmHg). How does using the actual pressure drop affect your calculations in part (a) (both qualitatively and quantitatively)?

Fig. 8.65

Effect of lung volume on pulmonary vascular (blood flow) resistance (From an animal lobe preparation). (Based on [69].) For Problem 8.22

8.22

(a) The pulmonary vascular resistance changes with lung volume. Figure 8.65 shows that it increases much with larger lung volumes, in part because the larger alveoli stretch the pulmonary capillaries. It also increases at very small lung volumes because these capillaries surrounding the alveoli become narrow. Calculate the range of pulmonary blood flow rates (in L/min), assuming this range of resistances and assuming that the pulmonary pressures are the same as in Problem 8.21.

(b) The pulmonary pressures actually change with lung volume. If they changed in a manner to keep the average flow the same as it is for a 110 mL lung volume, determine this change. (For the purpose of this calculation, assume that the ratio of Systolic pressuresystolic and Diastolic pressurediastolic pressures remains a constant and that the left atrium pressure remains the same.)

8.23

(a) Determine the overall compliances of the systemic arterial and venous systems by using Fig. 8.25.

(b) Is the ratio of these two compliances reasonable, given our model for compliance and the data for the vessels in both groups? (Consider only the large vessels in both groups.)

8.24

Does Fig. 8.25 suggest that sympathetic stimulation and inhibition change the vessels’ compliances, dead volumes, or both?

8.25

Compliance DistensibilityRelate the distensibility (introduced in Chap. 7) to $C_{\mathrm {flow}}$, $c_{\mathrm {flow}}$ and to $C^{\prime }_{\mathrm {flow}}$.

8.26

Poiseuille’s LawShow that the correction term to (8.19) is $(c_{\mathrm {flow}}^{2}/3 A_{\mathrm {d}}^{2})((P(0))^{2} + P(0)P(L) + (P(L))^{2})$.

8.27

(a) Poiseuille’s LawShow that (8.20) can be obtained from (8.19).

(b) Show that it can also be derived by integration of (8.14), expressed in r, using (8.7).

8.28

(a) Poiseuille’s LawShow that the correction term to (8.20), obtained by integrating (8.14) expressed in r and using (8.7), is $(1/2)(C^{\prime }{\mathrm {flow}}^{2}/r_{\mathrm {d}}^{2})((P(0))^{2} + P(0)P(L) + (P(L))^{2})$.

(b) Show that when the correction determined from the area approach (Problem 8.26) is expressed in terms of r, it becomes $(1/3)(C^{\prime }{\mathrm {flow}}^{2}/r_{\mathrm {d}}^{2})((P(0))^{2} + P(0)P(L) + (P(L))^{2})$.

(c) Why does this differ from the result in (a)?

Flow and Pressure

8.29

What is the average time blood spends in a capillary?

8.30

The cardiac index is the cardiac output divided by the person’s surface area. It normally ranges from 2.6 to 4.2 (L/min)/m$^{2}$. Use this to determine the cardiac output of a standard human. How does this value compare to the normal cardiac output we have assumed?

8.31

An artery with radius $r_{1}$ and blood speed $u_{1}$ divides into n arteries of equal radius. Find the radius $r_{2}$ and blood speed $u_{2}$ in these daughter arteries assuming that the pressure drop per unit distance $\mathrm{d}P/\mathrm{d}x$ is the same in the initial artery and each daughter artery.

8.32

Four veins with radius $r_{1}$ and flow speed $u_{1}$ combine to form one vein with radius $r_{2}=4r_{1}$. Find the flow speed in the larger vein.

8.33

The radius of each section of the aorta and the external iliac, femoral, subclavian, and brachial arteries each tapers as $\exp (-kx)$, where k is the tapering factor and x the distance along that vessel. Use Table 8.4 to find the tapering factor k for each vessel [57].

8.34

The design of blood vessels is sometimes optimized by minimizing a “cost function,” F, which is the sum of the rate work is done on the blood in the vessel and the rate that energy is used by metabolism through the blood in the vessel [24]. The first term is $Q(\Delta P)$, for flow rate Q and pressure drop $\Delta P$, and the second term is assumed to be proportional to the volume of the vessel of radius r and length L, $K \pi r^{2}L$, where K is a constant. Consequently, the cost function can be written as

$$\begin{aligned} F=Q(\Delta P) + K \pi r^{2}L. \end{aligned}$$

(8.114)

(a) For a resistive vessel, show that this becomes

$$\begin{aligned} F=\frac{8\eta L}{\pi r^{4}} \; Q^{2} + K \pi r^{2}L. \end{aligned}$$

(8.115)

(b) For a fixed vessel length and flow rate, show the optimal radius is

$$\begin{aligned} r_{\mathrm {opt}}=\left( \frac{16\eta }{\pi ^{2}K} \right) ^{1/6} Q^{1/3} \end{aligned}$$

(8.116)

and the minimum cost function is

$$\begin{aligned} F_{\mathrm {min}}=\frac{3\pi }{2} KLr_{\mathrm {opt}}^{2}. \end{aligned}$$

(8.117)

Fig. 8.66

a A planarBifurcated flow, bifurcating vessel. Determining variations in the length of each vessel for small planar displacements of B to $B^{\prime }$ in the b AB, c AC, and d DB directions (Based on [24].) For Problem 8.35

8.35

Bifurcated flow(advanced problem) We will use the cost function defined in Problem 8.34 to optimize the flow in a vessel of radius $r_{0}$ and length $L_{0}$ with flow rate $Q_{0}$, that bifurcates into a vessel of radius $r_{1}$ and length $L_{1}$ with flow rate $Q_{1}$ at an angle $\theta$ to the first vessel and one with radius $r_{2}$ and length $L_{2}$ with flow rate $Q_{2}$ at an angle $\phi$ to the first vessel, as seen in Fig. 8.66a [24]. Having straight, coplanar vessels minimizes the vessel lengths.

(a) Show that the total cost function is

$$\begin{aligned} F_{\mathrm {min}}=\frac{3\pi K}{2} (r_{0}^{2}L_{0}+r_{1}^{2}L_{1}+r_{2}^{2}L_{2}). \end{aligned}$$

(8.118)

(b) We can optimize the lengths and angles of the vessels by considering how the displacements of point B in Fig. 8.66b–d change the cost function. Show that any such movement of point B causes length changes $\delta L_{0}$, $\delta L_{1}$, and $\delta L_{2}$ that lead to a change of the cost function of

$$\begin{aligned} \delta F_{\mathrm {min}}=\frac{3\pi K}{2} (r_{0}^{2} (\delta L_{0})+r_{1}^{2} (\delta L_{1})+r_{2}^{2} (\delta L_{2})). \end{aligned}$$

(8.119)

This is optimized by setting $\delta F_{\mathrm {min,opt}}=0$.

(c) Show that moving point B along AB to $B^{\prime }$ by a distance $\delta$ as shown in Fig. 8.66b gives $\delta L_{0}= \delta$, $\delta L_{1}= -\delta \cos \theta$, and $\delta L_{2}= -\delta \cos \phi$, and $\delta F_{\mathrm {min,opt}}=(3\pi K \delta /2) (r_{0}^{2} - r_{1}^{2} \cos \theta - r_{2}^{2} \cos \phi ) =0$, and so it is optimized by

$$\begin{aligned} r_{0}^{2} = r_{1}^{2} \cos \theta + r_{2}^{2} \cos \phi . \end{aligned}$$

(8.120)

(d) Show that moving point B along CB to $B^{\prime }$ by a distance $\delta$ as shown in Fig. 8.66c gives $\delta L_{0}= -\delta \cos \theta$, $\delta L_{1}= \delta$, and $\delta L_{2}= \delta \cos (\theta +\phi )$, and $\delta F_{\mathrm {min,opt}}=(3\pi K \delta /2) (-r_{0}^{2}\cos \theta \,+\, r_{1}^{2}\, +\, r_{2}^{2} \cos (\theta \,+\,\phi )) =0$, and so it is optimized by

$$\begin{aligned} -r_{0}^{2}\cos \theta + r_{1}^{2} + r_{2}^{2} \cos (\theta +\phi )=0. \end{aligned}$$

(8.121)

(e) Show that moving point B along DB to $B^{\prime }$ by a distance $\delta$ as shown in Fig. 8.66d gives the optimization condition

$$\begin{aligned} -r_{0}^{2}\cos \phi + r_{1}^{2} \cos (\theta +\phi ) + r_{2}^{2}=0. \end{aligned}$$

(8.122)

(Note the symmetry in the last two equations.)

(f) Show that (8.120)–(8.122) can be solved to give

$$\begin{aligned} \cos \theta= & {} \frac{r_{0}^{4}+ r_{1}^{4}- r_{2}^{4}}{2 r_{0}^{2} r_{1}^{2}},\end{aligned}$$

(8.123)

$$\begin{aligned} \cos \phi= & {} \frac{r_{0}^{4}- r_{1}^{4} + r_{2}^{4}}{2 r_{0}^{2} r_{2}^{2}}, \end{aligned}$$

(8.124)

and

$$\begin{aligned} \cos (\theta + \phi )=\frac{r_{0}^{4} - r_{1}^{4} - r_{2}^{4}}{2 r_{1}^{2} r_{2}^{2}}. \end{aligned}$$

(8.125)

(g) Use continuity of flow and (8.116) to show that

$$\begin{aligned} r_{0}^{3} = r_{1}^{3} + r_{2}^{3}. \end{aligned}$$

(8.126)

(h) Show that (8.123) then becomes

$$\begin{aligned} \cos \theta =\frac{r_{0}^{4}+ r_{1}^{4}- (r_{0}^{3} - r_{1}^{3})^{4/3}}{2 r_{0}^{2} r_{1}^{2}} \end{aligned}$$

(8.127)

and find the analogous relations for (8.124) and (8.125).

8.36

Bifurcated flowUse Problem 8.35 to show that for optimized bifurcating vessels if [24]

• (a) $r_{1}=r_{2}$, then $\theta =\phi$,

• (b) $r_{2}>r_{1}$, then $\theta > \phi$,

• (c) $r_{2}$ is much greater than $r_{1}$, then $r_{2}$ approaches $r_{0}$ and $\phi$ approaches $\pi /2$,

• (d) $r_{1}=r_{2}$, then $r_{1}/r_{0}=2^{-1/3}=0.794 =\cos \theta$ and so $\theta = 37.5^{\circ }$.

These results generally agree with observations.

8.37

Bifurcated flowUse Problem 8.36d to show that it would take $\sim$30 generations of symmetric bifurcations starting with a vessel with the aorta radius of 1.5 cm to arrive at a vessel with the capillary radius of $5 \times 10^{-4}$ cm [24]. (Note, however, that such arterial divisions are usually not simple symmetric bifurcations.)

8.38

Allometric rules Diseases and disorders Law of Laplace Aneurysms Fusiform aneurysmsThere is a fusiform aneurysm in an aorta where the internal radius increases from $r_{1} ({=}1.25$ cm) in the normal section to $r_{2} = 1.3r_{1}$ in the diseased section, while staying at the same vertical height. The speed of blood flow is $v_{1} = 0.4$ m/s in the normal section and the (gauge) pressure $P_{1}$ is 100 mmHg. The blood density is 1,060 kg/m$^{3}$.

(a) Find the speed of blood flow $v_{2}$ in the aneurysm.

(b) Find the pressure $P_{2}$ in the aneurysm.

(c) Use the Law of Laplace to find the tensions required in the normal part of the aorta and in the aneurysm to maintain the pressure difference (from inside to outside the vessel). Compare these values.

(d) Describe how this increase in the tension needed in the aneurysm wall and the decreased strength of the wall (due to the thinner aorta wall in the aneurysm) can lead to an unstable situation.

8.39

Diseases and disordersThe normal inner radius of a large artery is 2 mm. It is 75 cm long, and the flow through it is 1/200 of the total blood flow. How would the pressure drop across it change if the flow through it were unchanged and there were severe stenosis in the artery

(a) across its entire length or

(b) across 5 cm of it?

(c) Alternately for (a) and (b), if the pressure at the beginning of the artery were 75 mmHg, would the pressure drop be severe enough to affect flow in the arterioles and capillaries?

(d) Alternately for (a) and (b), what added pressure would be needed at the beginning of the artery to maintain an unchanged flow in these arterioles and capillaries?

8.40

Diseases and disorders Stroke s@StrokesArteriosclerotic plaque narrows down a section of an artery to 20% of its normal cross-sectional area. What is the pressure in that section if immediately before it the pressure is 100 mmHg and the flow speed is 0.12 m/s?

8.41

The osmotic pressure of blood is 25 mmHg higher than that of interstitial fluid because it has a higher density of proteins. What is the difference in their densities of proteins that accounts for this?

8.42

(advanced problem) The blood Hematocrithematocrit is usually higher nearer the center of a blood vessel than at the blood vessel wall and has a distribution that we will take as $h(r)=H[1-(r/R)^{2}]$ from $r=0$ to R [51]. (The reason why flowing suspended particles, such as red blood cells, have higher concentrations near the center, called the Fahraeus–Lindquist effect, is not obvious.)

(a) The volume flow of a cylindrical shell in the vessel is $2\pi rv(r)\mathrm{d}r$, where $v(r)=2u\left( 1-r^{2}/R^{2}\right)$ from (7.40), so this flow weighted for the hematocrit is $2\pi rh(r)v(r)\mathrm{d}r$. Therefore the average hematocrit in the transported blood is $h_{\mathrm {av} }=\int _{0}^{R}2\pi rh(r)v(r)\mathrm{d}r/\int _{0}^{R}2\pi rv(r)\mathrm{d}r$. Show that $H=3h_{ \mathrm {av}}/2$.

(b) Now find the average value of the hematocrit at any given time in the blood vessel by calculating $\int _{0}^{R}2\pi rh(r)\mathrm{d}r/\int _{0}^{R}2\pi r\,\mathrm {d}r$. Show that for the parabolic distribution of hematocrit this volume-averaged hematocrit is $3h_{\mathrm {av}}/4$.

(c) The result in (b) states that the average hematocrit of the blood in the vessel at any given time is less than that in the blood that is being transported. Is this a contradiction?

8.43

(advanced problem) Repeat parts (a) and (b) in Problem 8.42 assuming $h(r)=H\exp (-r/R)$ and show the volume-averaged hematocrit is $0.88h_{\mathrm {av}}$ [51].

8.44

(advanced problem) The analysis in Problems 8.42 and 8.43 assumed that the parabolic v(r) we derived earlier assuming a constant viscosity is still valid when the hematocrit—and consequently the viscosity—decreases with radius. This should not be true. Qualitatively, how would you expect the spatially varying hematocrit and viscosity to affect the parabolic flow rate?

The Heart and Circulation

8.45

Would you expect cardiac muscle to be most similar to Type I, IIA, or IIB skeletal muscle? Why?

8.46

Someone wants to donate two pints of blood instead of the usual (and allowed maximum of) one. What consequences could this have?

8.47

AgingThe cardiac output of a woman remains at 5 L as she ages from 25 to 65 years of age, while her blood pressure increases in the average way.

(a) How does her total vascular resistance change?

(b) What fractional changes in vessel radius do this correspond to? (Assume typical conditions for arterioles.)

8.48

The volumetric flow rate out of a ventricle has been described in terms of the heart rate F and stroke volume $V_{\mathrm {stroke}}$ as $Q=FV_{\mathrm {stroke}}$, while flow rates have also been expressed in terms of the vessel cross section A and flow speed u as $Q=Au$. Explain why these two characterizations are either consistent or inconsistent.

8.49

If the cardiac output is 5 L/min and heart rate is 1 Hz, determine the volume of the left ventricle at its peak if the ejection fraction is 65%.

8.50

(a) If the inner volume of the left ventricle is 100 cm$^{3}$ and the wall volume is 30 cm$^{3}$, find the inner radius, outer radius, and wall thickness for the ventricle modeled as a hemispherical shell.

(b) Find the wall stress during systole.

8.51

When blood is pumped out of the left ventricle, it travels “upward” a distance of about 10 cm in the aorta during the $\sim$0.2 s duration of the peak of systole, stretching the walls of this very compliant vessel. There are no external forces on the body during this time, so the center of mass of the body does not move. Consequently, when the stroke volume of blood (of mass $m_{\mathrm {blood}}\simeq 70$ g) is ejected upward, the rest of the body (of mass $m_{\mathrm {rest}}\simeq 70$ kg) moves “downward” (ignoring gravity and frictional forces). This is the basis of the diagnostic method called ballistocardiography, in which a person rests horizontally on a light, very low friction horizontal suspension [4]. (Such devices have been used to develop methods that assess heart function, but are not in clinical use.) Assume the person is lying along the x direction on this “couch”—with his head pointing in the positive direction—and the center of mass of ejected blood is at $x_{\mathrm {blood}}$, that of the rest of the body is at $x_{\mathrm {rest}}$, and that of the entire body is $x_{\mathrm {body}}$.

(a) Show that $x_{\mathrm {rest}}=(x_{\mathrm {blood}}m_{\mathrm {blood}}+m_{\mathrm { rest}}x_{\mathrm {rest}})/(m_{\mathrm {blood}}+m_{\mathrm {rest}})$ always.

(b) Now let us call the positions in (a) those before systole. At the end of the main part of systole, the blood and rest of the body have moved by $\Delta x_{\mathrm {blood}}$ and $\Delta x_{\mathrm {rest}}$, respectively. Show that the body has moved by $\Delta x_{\mathrm {rest}}=-(m_{\mathrm {blood}}/m_{ \mathrm {rest}})\Delta x_{\mathrm {blood}}$ and that this is $-0.1$ mm.

(c) Because the blood moves with a constant velocity in this motion in the aorta, show that the velocity of the body during systole is $-0.5$ mm/s in the x direction.

(d) What is the average of $\Delta x_{\mathrm {rest}}$ during a full cardiac cycle? Why?

8.52

Compare the total mechanical and metabolic powers needed by the left heart and the right heart to pump blood.

8.53

Determine all the pressures, volumes, and flow rates in the overall body circulation model using the data provided in the Table 8.8. Do your answers agree with your expectations, such as the values in Table 8.1?

8.54

Pregnancy Fetus Ductus arteriosus Foramen ovaleDraw a diagram for the fetal circulation model, along the lines of Fig. 8.52.

8.55

Pregnancy Fetus Ductus arteriosus Foramen ovaleDerive and explain the equations of blood flow rates in a fetus (8.83)–(8.86).

8.56

(a) SumPregnancy Fetus Ductus arteriosus Foramen ovale the four equations of blood flow rates in a fetus (8.83)–(8.86) and show they are an identity [34].

(b) Explain why this means there are only three independent flow rate equations.

8.57

Pregnancy Fetus Ductus arteriosus Foramen ovale(a) Show that the fetal circulation model, with the four stated assumptions and equations (8.83)–(8.86), leads to the following solutions of the blood flow (with $KP_{\mathrm {v}}= Q_{\mathrm {R}}= Q_{\mathrm {L}}=Q$) [34]:

$$\begin{aligned} Q_{\mathrm {d}}/Q=(R_{\mathrm {p}}-R_{\mathrm {s}})/(R_{\mathrm {p}}+R_{\mathrm {s}}) \end{aligned}$$

(8.128)

$$\begin{aligned} Q_{\mathrm {f}}/Q=(R_{\mathrm {p}}-R_{\mathrm {s}})/(R_{\mathrm {p}}+R_{\mathrm {s}}) \end{aligned}$$

(8.129)

$$\begin{aligned} Q_{\mathrm {s}}/Q=2R_{\mathrm {p}}/(R_{\mathrm {p}}+R_{\mathrm {s}}) \end{aligned}$$

(8.130)

$$\begin{aligned} Q_{\mathrm {p}}/Q=2R_{\mathrm {s}}/(R_{\mathrm {p}}+R_{\mathrm {s}}); \end{aligned}$$

(8.131)

(b) Show that this means the foramen ovale is open, as assumed, for $R_{\mathrm {p}} > R_{\mathrm {s}}$.

8.58

Pregnancy Fetus Ductus arteriosus Foramen ovaleShow that closing the lungs totally in the fetal blood circulation models, by using $R_{\mathrm {p}} = \infty$, gives the limiting solution with $Q_{\mathrm {f}}=Q_{\mathrm {d}}=Q$, $Q_{\mathrm {s}}=2Q$, and $Q_{\mathrm {p}}=0$ [34], with Q defined as in Problem 8.57.

8.59

Pregnancy Fetus Ductus arteriosus Foramen ovaleImmediately after birth, the left and right hearts are still the same size (so $K_{\mathrm {R}} = K_{\mathrm {L}}= K$, the fetal circulation model assumption (2)) and the ductus arteriosus is still open ($R_{\mathrm {d}}=0$, assumption (3)), but now the lungs are inflated, so $R_{\mathrm {p}} < R_{\mathrm {s}}$, and the foramen ovale is now assumed to be closed. Show there is a self-consistent solution with a closed foramen ovale, and find $Q_{\mathrm {d}}$ and show that it is now negative [34].

8.60

Diseases and disordersThere is a hole in the septum that separates the left ventricle and right ventricle (Fig. 8.62).

(a) Explain why you would expect the pressure in the left ventricle to decrease and that in the right ventricle to increase.

(b) Explain why you would expect the stroke volume from the left ventricle to decrease because of this.

(c) Explain why the oxygenation of the blood in the left ventricle would decrease and that in the right ventricle would increase.

(d) If during systole the (gauge) pressure, stroke volume, and oxygenation levels (relative to that in the vena cavae) in the left ventricle each decreases by 10% as a result of this, explain quantitatively how the body could try to compensate for this?

8.61

Sketch the pressure-volume relationship, as in Fig. 8.58, for a disorder of the left ventricle that raises the overall pressure-volume curve at the end of systole and decreases systolic compliance. Describe how this affects stroke volume, cardiac output, and blood pressure.

8.62

(advanced problem)

(a) Solve (8.98) assuming that the flow $Q_{\mathrm {L}}(t)$ is a constant a from $t=0$ to $\alpha T$, and 0 from $t=\alpha T$ to T, where $0 \le \alpha \le 1$. (This repeats for every heart beat.) Note that the pressure at the beginning and end of each cardiac cycle is $P_{\mathrm {diastole}}$ and it becomes $P_{\mathrm {systole}}$ at $t=\alpha T$. (Hint: The analysis is similar to that for exciting an isometric muscle in Chap. 5 (see (5.8)–(5.10)) and temperature regulation in Chap. 13 (see (13.18)); also see Appendix C.)

(b) Show that $a=V_{\mathrm {stroke}}/\alpha T C_{\mathrm {sa}}$.

(c) Show the solutions from (a) lead to the relations

$$\begin{aligned} P_{\mathrm {diastole}} =P_{\mathrm {systole}} \exp (-(1-\alpha ) T/ \tau ) \qquad \mathrm {and} \end{aligned}$$

(8.132)

$$\begin{aligned} P_{\mathrm {systole}} = V_{\mathrm {stroke}}\tau /\alpha T C_{\mathrm {sa}} + (P_{\mathrm {diastole}}- V_{\mathrm {stroke}}\tau /\alpha T C_{\mathrm {sa}} ) \exp (-\alpha T/ \tau ). \end{aligned}$$

(8.133)

(d) Sketch $P_{\mathrm {sa}}(t)$ for several heart beats for $\alpha = 1/3$. Compare this sketch with those from the simple model in Fig. 8.55 and the real pulse in Fig. 8.53. Is this model better? Why?

(e) Show that when $\alpha = 0$ the solutions in (a) and (c) give the results presented in the text for the simpler model.

8.63

The solution to the classic Windkessel Model for steady-state flow that is suddenly turned off is exponential decay of the flow, as we saw in the simple model of the arterial pulse. In the electrical analog in Fig. 8.56 this corresponds to tracking the current when a constant voltage is initially applied and is suddenly turned off. Analyze this electrical problem analog and show that it has the same solution as the flow problem.

8.64

(advanced problem) Solve the electrical analog in Problem 8.63 for the three-element Windkessel Model.

8.65

(advanced problem) Solve the electrical analog in Problem 8.63 for the viscoelastic Windkessel Model.

Scaling

8.66

Allometric rulesCalculate the heart mass and heart beat rate (in beats per minute) for a man (70 kg), woman (50 kg), and an infant (5 kg) using the allometric relation parameters in Table 1.13.

8.67

Allometric rulesThe heart rate of mammals F is known to decrease with body mass as $m_{\mathrm {b}}^{-1/3}$. This seems to be true interspecies and also within a species. The human heart rate is known to decrease from infancy, through childhood and to maturity in a manner described better by body mass than age. Derive this relation using the dimensional analysis methods presented in Chap. 1. Assume that the stroke volume scales as body mass. Assume that a primary function of circulation is to bring warm blood from the core to the body surface to keep it warm. This means that the total blood flow rate scales as the rate of heat loss from the body.